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Mirrors > Home > MPE Home > Th. List > imass1 | Structured version Visualization version GIF version |
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.) |
Ref | Expression |
---|---|
imass1 | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssres 6008 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) | |
2 | rnss 5938 | . . 3 ⊢ ((𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶) → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶)) |
4 | df-ima 5689 | . 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
5 | df-ima 5689 | . 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
6 | 3, 4, 5 | 3sstr4g 4027 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3948 ran crn 5677 ↾ cres 5678 “ cima 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 |
This theorem is referenced by: predrelss 6338 vdwnnlem1 16935 dprdres 19946 imasnopn 23514 imasncld 23515 imasncls 23516 utoptop 24059 restutop 24062 ustuqtop3 24068 utopreg 24077 metustbl 24395 imadifxp 32265 esum2d 33555 eulerpartlemmf 33838 bj-imdirco 36535 brtrclfv2 42941 frege97d 42966 frege109d 42971 frege131d 42978 hess 42994 resimass 44402 setrecsss 47908 |
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